Refraction! Really? Refraction is the change in direction of a wave due to impedance changes in a medium or between mediums. For acoustic waves in the atmosphere refraction is typically be due to a temperature lapse (decreasing temp with increasing altitude) or temperature inversion (increasing temp with increasing altitude), which will cause an acoustic wave to bend upward or downward. It will not cause the wave to bend vertically (front to back). Reflection dwarfs refraction in the context we are discussing and it will never cause the wave to “wrap around” the motorcycle.

Your statement about point sources and parabolic reflectors is the first semi-correct statement you have made. Given a point source I can direct a wave in any direction I desire by manipulating the reflector. I can also widen or narrow its beam and manipulate its amplitude (again, search EIRP).

It seems that the fundamental mistake you are making is that you believe the source of the acoustic wave is at the opening of the motorcycle’s exhaust pipe. This is wrong. The source of the wave is the combustion that takes place in the engine. For the purpose of this discussion we will consider the combustion to be a point source that radiates an acoustic wave equally in all directions. Ahh, now we introduce the reflector…the combustion chamber acts as a reflector, directing the wave in one direction; through the exhaust valve. The wave then enters the exhaust pipe which acts as a waveguide. As the exhaust (an acoustic wave) exits the pipe it is directed toward the rear of the bike.

As mentioned earlier, acoustic waves in air travel by compression and rarefaction of the air molecules in the wave's direction of travel. Put your hand near to the opening of the exhaust pipe. You can feel the acoustic wave (the exhaust). As you move your hand further from the pipe opening the intensity of the wave decreases. If you place your hand to the side of the pipe, adjacent to the opening, you will not feel an acoustic wave (take care not to touch the pipe and confuse the vibration of the pipe with the acoustic wave).

In short, the acoustic wave travels in a conical straight line from the exhaust pipe, to the the rear of the bike. There is no magical property causing the wave to “make a U-turn or wrap around” the bike. If you are observing the bike while you and the bike are stationary you will hear the loud exhaust while standing in front of the bike. This is due to reflections (NOT refraction) of the wave from objects near the bike. Once the bike is moving the effect of the reflections is minimized and the loud exhaust will be almost entirely noticeable only from the rear of the bike.

The argument that a loud bike is a safe bike is a load of crap, unless someone is going to run you over from behind. On the other hand someone may intentionally run you over from behind for having a loud, obnoxious bike.

Unbelievable.
You win. Everything you've said is technically correct and not applicable to our discussion. I'm guessing you're very popular amongst your staff.